The objects and arrows in the cochain, which we assume to be countable from now
on, can be described as follows. Following the steps in [1]}
we can show that \(\mathcal {L}^n(1)\)  is isomorphic to
\(K^{A^\ast _n}\) , where \(A^\ast _n\)  is the set of all words of length less than
\(n\) , by induction on \(n\)  as follows. For \(n=1\) , \(\mathcal {L}(1)= K \times 1^A \cong K\cong K^{\lbrace \epsilon \rbrace } = K^{A^\ast _1}\)  by definition. By the induction
hypothesis, we have that an element \(\langle k,\sigma \rangle \in \mathcal {L}^{n+1}(1) =K \times \mathcal {L}^n(1)^A \cong K \times (K^{A^\ast _n})^A \cong K^{1+A\times A^\ast _n}\)  can be seen as a function that maps \(\epsilon \)  to \(k\)  and \(aw\)  to
\(\sigma (a)(w)\)  for \(a \in A\)  and \(w \in A^\ast _n\) . Similarly, it can be shown
that the arrow \(\mathcal {L}^n (!): \mathcal {L}^{n+1}(1) \rightarrow \mathcal {L}^n(1)\)  maps a function \(\sigma \in K^{A^\ast _n}\)  to its restriction \(\sigma \upharpoonright A^\ast _n\)  to the
subset of words of length less than \(n\) . We denote \(\sigma \upharpoonright A^\ast _n\)  by \(\sigma \upharpoonright n\)  from now on. Moreover, the limit of this
\(\mathsf {SMod}\) -cochain is \(K^{A^\ast }\)  together with the maps \(\zeta _n :K^{A^\ast } \rightarrow \mathcal {L}^n(1)\)  that sends a weighted language \(\sigma : A^\ast \rightarrow K\) 
to its restriction \(\sigma \upharpoonright n\)  to words of length less than \(n\) . The
limit cone is shown in the commuting diagram below.
\(\begin{tikzcd}& & & K^{A^\ast } [dlll, bend right=20, "\zeta _0"^{\prime }][dll, bend right=15, "\zeta _1"^{\prime }][dl, bend right=10, "\zeta _2"^{\prime }] \\1 [leftarrow,"!"]{r} & \mathcal {L}(1) [leftarrow,"\mathcal {L}(!)"]{r}& \mathcal {L}^2 (1) [leftarrow,"\mathcal {L}^2 (!)"]{r} & \ldots \end{tikzcd}\) 
